The story is unwieldy. The evening seems to end three times. A central character is not fleshed out sufficiently.

And yet, there’s something authentically endearing about Mosaic Theater Company’s “Paper Dolls,” the crowd-pleasing play with music at the Atlas Performing Arts Center. Buttressed by the Paper Dolls themselves — five Filipino health aides in Israel who moonlight in what they see as their true calling, as drag performers — the show places an entertainingly offbeat spotlight on marginal lives in a society ambivalent about their presence.

The Israel of 2004, where playwright Philip Himberg sets the action, proves to be a time and place of rich irony for the tale of these young men baring their not-so-secret souls. As played by Evan D’Angeles, Ariel Felix, Jon Norman Schneider, Rafael Sebastian and Kevin Shen, the Paper Dolls are escapees from destitution with whom it’s easy to empathize. And yet, the circumstances of their arrival in the Middle East are not morally neutral: They’ve been brought in to replace Palestinians as tensions and violence rose and the second intifada, or Palestinian uprising, commenced.

“Paper Dolls” is the account of these young men, sheltered from censure in their homeland, showing an increasingly religiously conservative Israel what freedom really means. The Dolls, whose talents do not seem to be consistently on a par with the singers they emulate, perform in seedy clubs, in cheap feather boas and dresses cut wittily from newspapers by designer Frank Labovitz. Their ultimate goal is never quite clear: International fame? A few extra shekels? A permanent visa? The play posits, though, that what they most fervently might be seeking is assurance that they can be themselves.

Director Mark Brokaw, whose lengthy list of fine off-Broadway productions includes the original stagings of Paula Vogel’s Pulitzer-winning “How I Learned to Drive” and Kenneth Lonergan’s “Lobby Hero,” guides the Paper Dolls to enjoyable embodiments of young men who just want to have fu-un, but also, bond profoundly with the beleaguered country. The musical numbers choreographed by J.M. Rebudal on James Kronzer’s well-conceived, two-tiered set, exude an exuberant, irreverent glee, especially in a wacky mash-up by the Dolls of “Lady Marmalade” and “Hava Nagila.” (The best number, though, is more shocking in its way, as the men are taken under the wing of a grubby club owner, persuasively played by Elan Zafir, who demands that they sully their act with Asian stereotypes.)

Brokaw has less success wrestling with a script that, despite a previous production at London’s Tricycle Theatre — Mosaic is presenting its U.S. premiere — still needs refinement. A subplot involving one of the health workers, Felix’s excellent Sally, who takes care of aged Chaim (the equally excellent Christopher Bloch), is a useful spine for the evening, and Lise Bruneau offers a compelling account of Chaim’s guilt-ridden daughter, Adina. But the frequent intrusions on the Dolls by the perplexing character of Etai (John Bambery), a young Israeli who claims to be making a movie of their exploits, seems intended to infuse the play with some added sexual ambiguity, and yet only prompts head-scratching.

 And while the Dolls’ gyrations are the production’s main event, some thought might be given to cutting a weaker number or two, especially toward the evening’s slightly — no pun intended — draggy end.

“Paper Dolls” is part of Mosaic’s Voices From a Changing Middle East festival, a years-long project of the company’s artistic director, Ari Roth. True to the festival’s mission, Himberg’s play takes the gag off muffled voices from that part of the world, and lets you hear them on this sweet evening, loud and clear.

Paper Dolls, by Philip Himberg, directed by Mark Brokaw. Choreography, J.M. Rebudal; music direction, William Knowles; set, James Kronzer; lighting, Brittany Shemuga; vocal arrangements, Howard Breitbart; costumes, Frank Labovitz; sound, David Lamont Wilson; projections, Sarah Tundermann. With Chris Daileader, Brice Guerriere, Dallas Milholland. About 2½ hours. $20-$65. Through April 29 at Atlas Performing Arts Center, 1333 H St. NE. Call 202-399-7993, Ext. 2 or visit mosaictheater.org.